Simplicial tree-decompositions of infinite graphs, I

This paper is intended as an introduction to the theory of simplicial decompositions of graphs. It presents, in a unified way, new results as well as some basic old ones (with new proofs). Its main result is a structure theorem for infinite graphs with a simplicial tree-decomposition into primes. The existence and uniqueness of such prime decompositions will be investigated in two subsequent papers. A simplicial decomposition of a graph is the recursively defined analogue to writing it as the union of two induced subgraphs overlapping in a complete graph, a ‘simplex’. These decompositions have successfully been applied in various branches of graph theory and elsewhere; a survey of such applications can be found in [ 2 ]. In a series of three papers we shall here consider the more theoretical aspects of simplicial decompositions. An overview of theoretical results (including those obtained in this series) and open problems concerning simplicial decompositions is given in [ 1 ]. If a graph has a simplicial decomposition into primes, i.e. into subgraphs that cannot be decomposed further, then these primes are essentially its smallest convex subgraphs. Unlike finite graphs, an infinite graph does not necessarily have a simplicial decomposition into primes, and if it does, this decomposition will not necessarily be unique. It is one of the oldest problems in the theory of simplicial decompositions to characterize the graphs that have a prime decomposition. In part two of this series [ 4 ] we shall obtain such a characterization for the simplicial decompositions of the most typical and common type, named ‘tree-decompositions’ after the shape in which their factors are arranged. (These simplicial tree-decompositions served as the prototype for the tree-decompositions recently introduced by Robertson and Seymour [ 13 ]). Our characterization of the graphs decomposable in this way is by a condition on the position of their separating simplices, a condition arising naturally from the structure of the known non-decomposable graphs. Part three of the series [ 5 ] deals with the uniqueness of simplicial tree-decompositions into primes: we shall prove that the uniqueness known for prime decompositions of finite graphs extends to simplicial tree-decompositions of infinite graphs, provided only that these are minimal in a certain very natural sense. In this paper, part one of the series, we give an introduction to simplicial decompositions and simplicial tree-decompositions, prove a basic theorem concerning their structure, and discuss some approaches to the problem of the existence of prime decompositions. We begin with some terminology. Let G be a graph, σ > 0 an ordinal, and let Bλ be an induced subgraph of G for every λ < σ. The family F = (Bλ)λ<σ is called a simplicial tree-decomposition of G (Fig. 1) if the following four conditions hold. (S1) G = ⋃ λ<σ Bλ. (S2) (⋃ λ<μ Bλ ) ∩Bμ =: Sμ is a complete graph for each μ (0 < μ < σ). (S3) No Sμ contains Bμ or any other Bλ (0 ≤ λ < μ < σ). (S4) Each Sμ is contained in Bλ for some λ < μ (μ < σ). If F satisfies (S1)–(S3) but not necessarily (S4), F is called a simplicial decomposition of G. If F satisfies (S1) and (S4), F is called a tree-decomposition of G. (This definition of a tree-decomposition is equivalent to, and in some cases more convenient than the original definition given by Robertson and Seymour [ 13 ]). If F = (Bλ)λ<σ is any fixed family of induced subgraphs of G satisfying (S1), and if H ⊂ G, x ∈ V (G) and μ ≤ σ, we denote by λ(H) the minimal λ for which Bλ ∩H 6= ∅,